by Chi-Fai Chan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 53-54). / Abstract also in Chinese. / Chapter Chapter 1 --- INTRODUCTION --- p.1 / Chapter §1.1 --- Polynomial Interpolation --- p.1 / Chapter §1.2 --- Conjugate Gradient Type Methods --- p.6 / Chapter §1.3 --- Outline of the Thesis --- p.10 / Chapter Chapter 2 --- INTEGRAL EQUATIONS --- p.11 / Chapter §2.1 --- Integral Equations --- p.11 / Chapter §2.2 --- Numerical Treatments of Second Kind Integral Equations --- p.15 / Chapter Chapter 3 --- FAST ALGORITHM FOR SECOND KIND INTEGRAL EQUATIONS --- p.20 / Chapter §3.1 --- Introduction --- p.20 / Chapter §3.2 --- The Approximation --- p.24 / Chapter §3.3 --- Error Analysis --- p.35 / Chapter §3.4 --- Numerical Examples --- p.40 / Chapter §3.5 --- Concluding Remarks --- p.51 / References --- p.53

Lui Man Leung. / "May 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

This paper is about the computation of the stresses on a rigid body from a knowledge of the far field velocities in exterior Stokes and Oseen flows. The surface of the body is assumed to be bounded and smooth, and the body is assumed to move with constant velocity. We give fundamental solutions and derive boundary integral equations for the stresses. As it turns out, these integral equations are singular, and their null space is spanned by the normal to the body. We then discretize the problem by replacing the body by an approximating polyhedron with triangular faces. Using a collocation method, each integral equation delivers a linear system. Since its matrix approximates a singular integral operator, the matrix is ill-conditioned, and the solution is unstable. However, since we know that the problem is uniquely solvable in the hyperspace orthogonal to the normal, we use regularization methods to get stable solutions and project them in the normal direction onto the hyperspace. / Graduation date: 1999

We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. The boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. To study the problem we apply a familiar method of Vekua-Muskhelishvili which consists in using a conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This latter is reduced in turn to a Toeplitz operator equation on the unit circle with symbol bearing discontinuities of second kind. We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions.

Graduation date: 1991

Integral equations form an important subject with applied mathematics due to their occurence in a variety of models of physical problems. The intent of this thesis is to present in a simple and concise manner the theory of integral equations in the context of their solution. A survey of the types of methods used for the approximation of linear integral equations is made along with the types of equations to which they may be applied. Detailed examples are presented for each of the methods discussed and wherever feasible, computer methods are employed.Ball State UniversityMuncie, IN 47306

The objective of the thesis is to treat the Schrodinger equation in parallel with a standard treatment of the heat equation. In the books of the Rubensteins and Kress, the heat equation initial value problem is converted into a Volterra integral equation of the second kind, and then the Picard algorithm is used to find the exact solution of the integral equation. Similarly, the Schrodinger equation boundary initial value problem can be turned into a Volterra integral equation. We follow the books of the Rubinsteins and Kress to show for the Schrodinger equation similar results to those for the heat equation. The thesis proves that the Schrodinger equation with a source function does indeed have a unique solution. The Poisson integral formula with the Schrodinger kernel is shown to hold in the Abel summable sense. The Green functions are introduced in order to obtain a representation for any function which satisfies the Schrodinger initial-boundary value problem. The Picard method of successive approximations is to be used to construct an approximate solution which should approach the exact Green function as n goes to infinity. To prove convergence, Volterra kernels are introduced in arbitrary Banach spaces, and the Volterra and General Volterra theorems are proved and used in order to show that the Neumann series for the L^1 kernel, the L^infinity kernel, the Hilbert-Schmidt kernel, the unitary kernel, and the WKB kernel converge to the exact Green function. In the WKB case, the solution of the Schrodinger equation is given in terms of classical paths; that is, the multiple scattering expansions are used to construct from, the action S, the quantum Green function. Then the interior Dirichlet problem is converted into a Volterra integral problem, and it is shown that Volterra integral equation with the quantum surface kernel can be solved by the method of successive approximations.

Thesis (doctoral)--Uppsala universitet, 1917. / Includes bibliographical references.

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (leaves 150-155).

published_or_final_version / Mathematics / Master / Master of Philosophy